LINKAGE EFFECTS ON ADDITIVE GENETIC VARIANCE AMONG HOMOZYGOUS LINES ARISING FROM THE CROSS BETWEEN TWO HOMOZYGOUS PARENTS

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1 LINKAGE EFFECTS ON ADDITIVE GENETIC VARIANCE AMONG HOMOZYGOUS LINES ARISING FROM THE CROSS BETWEEN TWO HOMOZYGOUS PARENTS W. D. HANSON AND B. I. HAYMAN Department of Genetics, North Carolina State College, Raleigh, North Carolina and Applied Mathematics Laboratory, Christchurch, New Zealand Received December 26, 1962 evaluation of linkage effects represents an area of theoretical genetic TEearch on quantitative genetic characterization where there exists a multitude of problems with few solutions. SCHNELL S work (1961, 1963) represented the first general solution of the effects of linkage on genetic variances estimated from mating designs imposed within genetic populations at linkage equilibrium. The problem is complicated when considering quantitative genetic studies in an organism such as a self-pollinated species where the initial cross is normally made between two homozygous parents, and the reference (F,) population is in linkage disequilibrium. MATHER S solution (1949) and the general solution of GATES, COMSTOCK, and ROBINSON (1957) considered linkage in this context, as did COMSTOCK and ROBINSON S ( 1952) evaluation of the mating system (defined as Design 111) for measuring degree of dominance. These three solutions required a pairwise evaluation of linked loci which is difficult to interpret in the context of n linked loci. This paper is designed to evaluate as a general concept the effect of linkage on genetic variance estimates. Inbred relatives from homozygous parents will be considered which represents the typical quantitative genetic study reported in self-pollinated crops ( HANSON 1961 ). The development considers random homozygous lines arising from the cross between two homozygous parents where m generations of intermating are introduced after the F, generation, prior to selfing. The base unit for inheritance will be the chromosome, and the number (n) of linked loci at which the two parents differ will be general. One of the objectives for this paper is to develop a procedure for describing the effects of linkage when the material is initially at linkage disequilibrium. The simplest case involves the variability among random homozygous lines from a cross between two3 homozygous lines, and is considered in this paper. The technique is being extended to the effects of linkage on the variability of family means involving homozygous lines and on the dominance estimates obtained from Design I11 of 1Published as Journal Paper No of the North Carolina Agricultural Experiment Station and supported in part by grants from the Rockefeller Foundation and the National! Institutes of Health. Genetics 48: June 1963.

2 756 W. D. HANSON AND B. I. HAYMAN COMSTOCK and ROBINSON ( 1952). Further, the description of linkage effects and the effects of intermating upon the genetic variability of material arising from two homozygous parents are of extreme interest to both quantitative geneticists and plant breeders working with self-pollinated or open-pollinated crops. THEORETICAL DEVELOPMENT Assumptions. Some simplifying assumptions are necessary for this development: (i) The loci at which two parents differ all contribute equally without epistasic interactions and are equally spaced with respect to the genetic map scale for a chromosome, and (ii) the occurrence of a favorable or unfavorable condition of a locus at a position within a progenitor chromosome is random, subject to the restriction that nq favorable loci are contributed by one parent and n (l-q) by the second. Gene frequencies are one half. The manner in which these assumptions may bias the results will be noted later. Chromosome effects. The system is initiated by the crossing of two homozygous parents giving two progenitor chromosome types with respect to a pair of homologues. Identify the homozygous types by b and c. The two homologues differ at n loci with nq favorable and n( 1-9) unfavorable genes being identified with chromosome type b. If the genetic values of a locus in the two phases are coded +1 and -1, the most favorable homozygous chromosome effect would be n and the most unfavorable homozygous chromosome effect would be -n. The genetic values of the parents are xb = [q- (I-q) ] n = (2q-1) n and x, -(2q-1) n. Genetic variance and linkage. The average recombination per chromosome has been defined by GRIFFING (1960) and extended for a mating system by HANSON (1962). Suppose that n loci are segregating in one linkage group and that after the mating procedure nb of these loci in a gamete are derived from one parental chromosome and n, of these loci from the other parental chromosome. The total number of linkage pairs is n(n-l)/2 and the number of pairs not in the initial linkage phase is nbn,. The average recombination is defined to be the proportion of new linkage pairs, p(n,nb) = 2nbn,/n(n-l). (1) The genetic value of a homozygous individual depends on the proportions of favorable and unfavorable loci contributed by each progenitor. Suppose that each progenitor contributes nb and n, loci respectively and that nb. and n,. of each set are favorable and nb2 and ncz unfavorable. Then the genetic value of the individual is X = nbl- nbz + n,, - ncz. Since nbl+ ncz = qn and nbz + nc1 = ( 1-q) n, X(n,nbl,nbz) = 2(nbl-nb2) - (2q-l)n. Given the average recombination p, the genetic value X will vary with the partitioning of nb into nbl favorable and nb2 unfavorable loci. Each p will be associated with a range of X. The nb loci of one progenitor type are a sample from a population of n loci divided into favorable and unfavorable in the propor-

3 LINKAGE AND VARIANCE 75 7 tion q: ( 1-4). The proportion of favorable loci in the nb loci follows a hypergeometric distribution so that E(nb1 I n7q7nb) = qnb and var(nbl I n,q,nb) = E(nb1- qnb)2 = q( 1-q) nbnc/(n-l). Since nbl + nb2 = n b, X = 4nb1-2nb - (2q-1)n =4(nbl--nb) + (2q-1)(212b-T2). Hence, E(X2 I n,q,nb) = 16E(nb1 - qnb) + (2q-1) (2121, - n)2 = 16q(l-q)nbnc/(n-l) + (29-1)*(nZ-4nbn,). Since from (1 ) nbnc = n (n-1) p (%nb)/2, E(X2 I n,q,nb) = n (2q-1)2(1-2p) + 2np, (2) where it is understood that p is an average over the same class of individuals as specified for the expectation of X2. Now E (X I n,q,nb) = (2q-1) (2nb-n), which is not generally zero so that E ( X2 I n,q,nb) is not a variance. In deriving the expectation of the average recohbination, HANSON ( 1962) considered firstly homologous chromosomes with given nb (y in that paper), secondly the wider class with a given number of recombinations z, and thirdly the still wider class with a Poisson distribution of z based on an average equivalent map length s. Equivalent map length was used to describe that hypothetical map length which would generate the distribution of z for a mating system in one meiotic cycle. Following the same sequence of widening classes, it is clear that so that E(nb I n,z) =E(nb I n,s ) = n/2 E(X I n,q,z) = E(X 1 n,q,s ) = 0. Hence from (2), var(x I n,q,z) =E(Xa I n,q,z) = n2(2q-1)2(1-2p) + 2np, var(x I n,q,s ) =E(Xa I n,q,s ) = n2(2q-1)2(1-2p) + 2np with the same understanding about the scope of p as noted for (2). At linkage equilibrium s is large and p = % (see (6) and (9)) so that var (X I n,q) = n. The relative genetic variance at disequilibrium is R(n,q,s ) = var(x I n,q,s )/var(x I n,q) = n(2q-l)2(1--2p) + 2p. (5) HANSON (1962, Table 3) gives values of p for N = n-1 and s, from which R(n,q,s ) may be calculated. The appendix contains the derivation of exact formulas for p(n,z) and p(n,s ). (3) (4) DISCUSSION Effects of parameters on R(n,q,s ). The relation (5) for the relative genetic variance may be written R (n,q,s ) = d( 1-2p) + 2p, (5a)

4 758 W. D. HANSON AND B. I. HAYMA?; where d = n(2q-l)?. R is a bilinear function of the two parameter5 d a u p ~ which are in turn functions of the four basic parameters: n, the number of loci on a chromosome; q, the proportion of favorable loci in the better parent; s, the genetic map length; and m, the number of generations of intermating. Different combinations of n, q, s, and m can produce the same value of R, and different combinations of n, 4, and s can produce the same trend in R with increasing generations of intermating, m. Conversely, a value of, or trend in, R does not permit an inference to the value of n, q, s, and m unless three of these parameters are already known from other sources. However, a consideration of the functional form of R can show how the parameters are combined, and if any special circumstances exist, in which inferences about the parameters are possible. Two of the parameters, s, and m, are always combined (HANSON 1959) into the equivalent map length s = s (mf4)/2. (6) Since s is fixed in an experiment, s varies directly with m, and s only influences the rate of change of s relative to m. The average recombination p is determined by n and s according to formula (Io) or HANSON S (1 962) Table 3. p increases with increasing s to a limit of one half (see (9) and (12)). It decreases with increasing n to a limit determined by s, and for n > 10, p is virtually independent of n and has the approximate value (formula 11 ) : p(s ) = (1-2s + 2s Z - e-z~ )/4s ~. Since in practice n is usually greater than ten, the one apparent disadvantage of (5a), that both d and p are influenced by n, is usually not important. The relative genetic variance (5a) contains two terms. The first term d( 1-2p) vanishes when q = or the initial parents have equal genetic values. It can be termed the main component of the genetic variance. The second term 2p arises from sampling the initial chromosomes. Both components are influenced by the parameters. From the previous discussion it is clear that the main component increases with increasing n and q but decreases with increasing m. The sampling component decreases with increasing n, is unaffected by q, and increases with increasing m. Total genetic variance evidently increases with increasing q, but the effects of n and m on the total genetic variance are not clear since they each influence the two components in opposite directions. Another way of writing (5) is R(n,q,s ) - 1 = (d-i) (1-2p). (5b) Since (1-2p) is positive, R is greater or less than unity according as d is greater or less than unity. The sole determiner here of whether relative genetic variance is above or below its value at linkage equilibrium is n(2q-l)2, and this is so whatever the map length or the extent of intermating. When q is within the range of 1/2 i. 1/2v ; or n < 1/(2q-l), the relative genetic variance is below its equilibrium value. When d increases with p held constant, then R increases and this confirms the increase in R with q noted above. When p increases with d held constant, R decreases or increases according as d is greater or less than one, and this must be

5 LINKAGE AND VARIANCE 759 the pattern of change in R with change in m. Figures 1 to 3 illustrate this change in relative genetic variance with continued intermating and for various values of q and n. As intermating proceeds, p tends to % and R approaches one either from above or below according to the sign of (d-1 ). This trend in genetic variance with increasing intermating seems to permit one inference about the parameters. If the variance increases, then n (2q-1) 2< 1, SO that either q is near % or the number of loci is small. Conversely, if two similar parents produce a large genetic variance which decreases on intermating n (2q-1) > 1 and the number of loci must be large. The effect of n on R is still complicated because it influences both d and p. If d>l, relation (5b) shows that R increases with n, and this is probably the usual situation in practice. However, when q = 1/, d = 0 and R(n,%,s') = 2p. Only the sampling component remains and this decreases with increasing n. When O<d<l, an intermediate situation may prevail, and R may decrease initially before finally increasing with n. Average R(n,q,s'). The complete lack of information about n, q, and s' in a study complicates the generalization which can be made. It would be desirable to M 3 \ , U) t lo.55 x) FIGURES 1-3.-Relative genetic variance among random homozygous lines R (n, g, s'), if m generations of intermating were imposed on the F, generation. FIGURE 1.-Given 11 loci equally spaced on a genetic map of 100 centimorgans in length and selected levels of 9 (.50, 50,.70,.80, and.go). FIGURE 2.-Given 50 loci equally spaced on a genetic map of 100 centimorgans in length and selected levels of 9 (.50,.55,.60,.65,.70, and.75). FIGURE 3.-Given 100 loci equally spaced on a genetic map of 100 centimorgans in length and selected levels of q (.50,.55, 60,.65, and.70).

6 760 W. D. HANSON AND B. I. HAYMAN have a reference for discussion. Consider that a population of homozygous genotypes for an improved species is available and that the population is at gene equilibrium with respect to the homozygous conditions for loci. Let the frequency of the (+) and (-) homozygous condition by U, and ut, respectively. Then the possible combinations with respect to the ith locus for randomly selected pairs of parents would be: Frequency: U, u,u, UlUl U:: Parent I - - Parent I1 + + The relative genetic variance for homozygous progeny (R(n,q,s ) ) involves only those loci at which the two parents differ. For the single locus case, the probabilities that parent I is (+) and 11, (-) and parent I, (-) and 11, (+) are equal. Similarly, for those pairs of parents which differ at exactly n loci, average q can be shown to be and average d to be one. Hence, E[R(n,q,s )] = (1-2p)E[d] + 2p = 1 for a given n and s, and E[q] = s. That is to say, if the two parents are random selections from the population, R (n,q,s ) may increase or decrease with increasing m (number of generations of intermating), but for the average of all experiments R (n,q,s ) would equal its equilibrium value and not change with increasing m. Further, these points would apply to families of homozygous lines arising as selfed progeny from random individuals within a population. The information can be summarized as follows: Approximate proportion Change in R ( n,q,s ) Range for q of experiments with increasing m 0 to (1/2-1/2dG) 1 /6 Decreases (1/2-1/2+2) to (1/2+1/2+) 2/3 Increases (1/2+1/2+) to /6 Decreases Thus, or experiments based on pairs of random homozygous parents, it would be difficult to demonstrate changes in R (n,q,s ) except for a limited proportion of experiments considerably beyond the range (1/2* 1/2\/n) for q where R(n,q,s ) changes markedly with increasing m. (See Figures 1 to 3.) Parents selected for quantitative genetic studies in self-pollinated species are normally members of the improved set of genotypes for a species. Estimates for genetic parameters must reflect the population structure of improved genotypes to have meaning in a breeding program. Parents are not, however, random genotypes of this improved set but usually selected to represent differences. The criterion for selection of parents may be phenotypic differences, or perhaps more frequently, differences in the parental backgrounds. Therefore, parents used for quantitative genetic studies should be less similar than randomly selected parents from the improved set, and genetic variances decrease with intermating for an average of experiments. However, genetic variance estimates may increase or decrease with intermating for specific experiments. The range within which experi-

7 LINKAGE AND VARIANCE 761 ments might fall is speculative; however, extreme cases would not be expected since the parents are normally selected from the set of improved genotypes. Studies which involve an adapted genotype and a primitive (or exotic) genotype would involve values for q near one (or zero) and extreme linkage effects would be expected. The quantity (9) has been defined as the proportion of desirable loci found in one parent and may take values between zero or one. Minimum frequency of coupling phase linkages with reference to the parents occur for q = with a maximum frequency at one or zero. The frequency of coupling phase linkages is [n-1-2nq( 1-q)]/(n-I) so that q or (l-q) have equivalent effects with respect to the concept of coupling, or as evident from previous formulations, with respect to the interpretation of genetic variances. Assumptions. The assumptions made for this development will now be examined. Two restrictions were that the locus effects were equal and that the loci were equally spaced with respect to the genetic map scale. The failure of these two restrictions to be met in practice would affect primarily the sampling variability, and the formulation presented would tend to underestimate this source of variability. If the number of loci per chromosome map should be large, these two restrictions would be of little consequence. A third restriction assumed that, in the progenitor chromosome, nq favorable and n( 1-q) unfavorable loci were distributed at random among the n loci. Relation (5) is only correct when averaged over parental chromosome pairs containing all these permutations and is not necessarily correct for a particular pair of progenitor chromosomes. Relation (2), however, is correct for a particular arrangement of favorable and unfavorable genes because the manner in which the nb loci can be divided into ab1 favorable and nb2 unfavorable loci is not restricted there by the number of points of recombination, z. When z is restricted as in (3), (4), and (5), nbl has a random hypergeometric distribution only if the sampling is considered to be the average of samplings with restricted z from a population of progenitor chromosomes with nq favorable loci distributed at random. HANSON (1959) has shown that the number of points of recombination follows a Poisson distribution. This was based on the assumption that a point of recombination occurs independently with a uniform distribution in the genetic map scale. The assumption of independence could be reasonable, specifically when one interposes several generations of intermating. A tacit assumption has been that the relative genetic variance deduced in (5) for a single chromosome is applicable to a complete genotype. More correctly, if there are k chromosomes, the relative genetic variance is k k R=.Z z=l rn:(2q*--1)2(1--2pl)+ 2ntpz1/.Z z=l nc k k = I: n,r(n,,q,,s,)/z 72% i= 1 i=l which is a weighted average of the relative genetic variances of the separate

8 762 W. D. HANSON AND B. I. HAYMAN chromosomes. For example, the trend in R with m would be a weighted average of several of the graphs in Figures 1 to 3. It would even be possible for R to be constant with progressive intermating without all the (d,-1) being zero if some (d,-1 ) were positive and some negative. However, if the chromosomes are similar in size and effect, the previous discussion applies to a complete genotype. Finally, nonadditive variability was ignored, or considered to be negligible. The complexity of the problem forced this simplification, the argument being that at least a concept of linkage effects on the genetic variability could be developed. When dealing with the variability among homozygous lines, only the nonadditive variability arising from the interaction of additive scales would be present; however, even with this simplification, the evaluation is extremely complex. Experimental ramifications. The disturbing conclusion from this study is that genetic studies involving homozygous lines from homozygous parents yield results with restricted interpretation. The interesting facet from a theoretical point of view is that different combinations of the extent of coupling phases in the original parents and the number of loci per genetic map produce similar modifications of genetic variances. Figures 1 to 3 were based on a map length of 100 centimorgans with loci at ten, two, and one intervals, respectively. From these figures and from the previous discussion on average R(n,q,s ) and q one would expect that for typical experiments the genetic variance among lines should decrease with intermating. A summarization of quantitative genetic information available from crosses between two homozygous parents is of interest: (i) The expected value for the genotypes with a given number of recombinations (as noted in (4) and by other workers) is zero and thus independent of the number of generations of intermating. Therefore, population means will not change with intermating unless there exist nonadditive effects associated with the change in linkage disequilibrium. (ii) Although genetic variability can be grossly inflated as compared to that expected for linkage equilibrium, the genetic variances reflect the potential progress a breeder could expect from selection among the homozygous progeny. The variances could be misleading for predicting the progress in the intercrossed progeny from these selected lines. (iii) The detection of changes in genetic variances with interposed generations of intermating established linkage between loci affecting a character; however, the failure to detect changes in genetic variance does not rule out the presence of linked loci. On the other hand, similar estimates for imposed generations of intermating would indicate estimates near the equilibrium value. (iv) Estimates of genetic variances available from crosses between pairs of homozygous parents of a species could yield a realistic characterization of the genetic variability for a population. The critical point would be whether one could assume that the crosses involved pairs of random parents, a condition which may not be met. The average for experiments could lead to an overestimation of equilibrium variances; however, this point is speculative. A potential danger exists when genetic variances estimated from cross between two homozygous parents are used to describe the dynamics of a population, which is the key to plant breeding.

9 LINKAGE AND VARIANCE 763 The ultimate goal in plant breeding programs is achieved only through the manipulation of populations. A program in self-pollinated crops does not differ from other crops in this respect but does introduce the limitation of an essentially closed system with respect to genetic recombinations. Genetic recombinations within selected sets are obtained by forced intermating. Descriptive effects of intermating with respect to lengths of linkage blocks are available (HANSON 1959). Intermating prior to selection would probably decrease genetic variability and thus immediate genetic progress. A distinction must be made between immediate gains from selection and real gains within a population. The approach to the idealized genotype will be achieved only through some recurrent selection program which permits recombination to occur under selection pressure. Whether a plant breeder wishes to intermate a generation or two before selfing and selecting as previously suggested by the author involves an evaluation of the material and of economics and time. Opportunity for recombination with segments of a population is essential to achieve the ultimate goal of plant breeding. Attention in this paper was directed towards quantitative genetic studies in self-pollinated crops where the lines arose from two homozygous parents. Studies which involved homozygous parents or two populations such as a Latin American corn variety and a corn belt variety, for example, have many problems in common with those discussed in this paper. SUMMARY The additive genetic variance among homozygous lines arising from a cross between two homozygous parents was related to their average recombination. The procedure was generalized to include intermating after the F, generation. To evaluate required certain simplifications. The characterization of changes in the genetic variances with intermating involved proportions of coupling-phase linkages (determined by q) in the original parents, and the number n of loci per genetic map of a chromosome which have interrelated effects. Genetic variances may be less than or considerably greater than that expected for linkage equilibrium according as n (2q- 1 ) is less than or greater than unity, but will always move towards the equilibrium value with intermating. When n (2q-I ) * is unity, the variances do not change with intermating. The unit for the variability analysis was the chromosome. The extension to total genotypic variability involved the summation for k pairs of homologous chromosomes. Changes in genetic variability with intermating established linkage effects; however, the converse would not be true. The average additive genetic variance for experiments which involved random pairs of homozygous parents would estimate the variance for equilibrium condition and not change with intermating. Unless one dealt with a cross between two extreme parents, it would be extremely difficult to establish changes in genetic variability with intermating. APPENDIX The expected recombination can be specified exactly in terms of n and z or of

10 764 W. D. HANSON AND B. I. HAYMAN n and s'. The derivation here parallels HANSON'S (1962) approximate derivation and assumes without restatement the same basic situation. Any homozygous individual arising from a cross has alternate homozygous segments of the two progenitor chromosome types. The numbers of loci in the two sets of alternate segments are nb and nc. Take expectations of (1) over all nb generated by a given number of recombinations x: The variance of nb determines the recombination value and its derivation is the main problem of this appendix. Consider the set of homologous homozygotes containing x points of genetic recombination within n = nb+nc loci. If these x are distributed at random, the frequency pattern of recombinations is generated by where N = n-1 and summation is over all a, such that a,+a,+a3+... E x. The coefficient of tlait2aatja3... is the chance that a, points of recombination lie between loci 1 and 2, u2 between loci 2 and 3, and so on. Let y be the number of loci in the same progenitor set as the first locus. Then locus if1 is in this set if the number of recombinations between it and the first locus is even and is in the other progenitor set if this number of recombinations is odd. This means that locus i+l contributes. [l + (-l)ql+",+...a%]/2 to the total y of loci. Hence, N y= 1 + z [l + (-1)%+%+...+5]/2. 2=1 Progenitor sets of loci which do not include the first locus contain N+l-y loci, and E(nb 1 N,x) =Ey/2 + E(N+l-y)/2 = (N+1)/2, as in (3). var(nb 1 N,s) =E[nb-(N+1)J2]2 = E[y- (N+ 1 )/e]* =E[1 (-l)"~~(-l)ui+a~~(-l)a~+a~+a~+..]'/4 = E[1 2(-1)%+2(-1)a*+a2+2(-1)a1+%+~, (-1)az +2(-1)%+a, (-1)" ]/4.

11 LINKAGE AND VARIANCE 765 Now E(-l)'~'ul'.. is obtained by placing t, = ti =.. = -1 and tk = 1 for k # i, j,.. in G(tl,t2,... tn). If there are r loci involved in the exponent, then E(-l)%'", " = N"(N-r-r)" (1-2r/N)", and this is independent of the particular loci involved. Hence, where var(n, I N,z) = [ Nfl + 2N(l-2/N)" + 2(N-l) (1-4/N)" f...1/4 N = (N4-1-r) (1-2r/N)"/2 - (N+1)/4 r=0 [Nh(N,x+l) + (N+2)h(NYz)- (N+1)]/4, N h(n,z) = X (l--er/n)x. r=0 Then from (7) with n = Nfl, p(n,z) = [(N+1) (Nf2)- Nh(N,z+l) - (N+2) h( N,x) ]/2N( N+ 1 ). (8) Note that h(n,2x+l) = 0 so that there are virtually two different formulas for p(n,z), one for even x and one for odd z. HANSON (1959) has shown that after m generations of intermating, z has a Poisson distribution with mean s' = s (nf4)/2 where s is the genetic map length of the chromosome. The average recombination for this distribution of x is m p (Np' ) = x p ( N,z) e-s's'x/x! x=o N = % - x (N+1-r)e-2~8'/y/N(N+I) (9) r=1 (N+1) (N+2)+ N(N+l)e4*'/H- 2N(N+2)e*s'/N- 2cz5' - 2N(N+1) (10) [ezs'/x- 112 Some alternative formulas are useful for computation. In the case of h(n,z) use the definitive N h(n,z) = r. (1-2r/N)" r=o for all x when N is small. Use -1) (N/2)--2c+l for all N when x is small and even. B, are the Bernoulli numbers, B, = 1/6, B, = 1/30, etc. An approximation which is correct for N = 1, 2,00, for z = 0, 2, CO, and for x = 0.8N = CO is (N-1) (N-2) h(n,z) =2 + (N+x-2) (z+l) Substitution of these formulas for h(n,z) in (8) gives formulas for p(n,z). In the approximate case these reduce to 1 1 (N-1) (N-2) (N+2),u(N,x) = -- -, for even x, 2 N(N+l) 2N(N+1) (Nfz-2) (zfl) = (N-I) (N-2), for odd x. 2 N(Nf1) 2(N+1) (N+x-1) (zf2)

12 766 W. D. HANSON AND B. I. HAYMAN These are correct at the boundary values of N and x and otherwise have errors of up to 3 percent. The asymptotic behavior of p (N,s') is given by the following approximations: For large N p(n,s') = ( 1-2s'+2s'~-e-~~')/4~'L-(3-l8s'2~8s'7-3e-"'-6s'e-?Y')/l2s'~~. (11) For large s' p(n,s')= 1/2 - e-2s'/"/(nfl). (12) LITERATURE CITED COMSTOCK, R. E., and H. F. ROBINSON, 1952 Estimation of average dominance of genes. Chapter 30. Heterosis. Edited by J. W. GOTEN. Iowa State 'College Press. Ames, Iowa. GATES, C. E., R. E. COMSTOCK, and H. F. ROBINSON, 1957 Generalized genetic variance and covariance formulae for self-fertilized crops assuming linkage. Genetics 42 : GRIFFING, G., : Accommodation of linkage in mass selection theory. Austral. J. Biol. Sci. HANSON, W. D., 1959 The breakup of initial linkage blocks under selected mating systems. Genetics 44: Resolution of genetic variability in self-pollinated species with an application to the soybean. Genetics 46: Average recombination per chromosome. Genetics 47: M7415. MATHER, K., 1949 Biometrical Genetics. Dover Publications, Inc. London. SCHNELL, F. W., 1961 Some general formulations of linkage effects in inbreeding. Genetics 46: The covariance between relatives in the presence of linkage. pp Stutisticul Genetics and Plant Breeding. Edited by W. D. HANSON and H. F. ROBINSON. Humphrey, New York.